#1279191

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

(2,3)(5,6,10,12)(7,8,11,9)(21,22)(23,28)(24,29)(25,30)(26,27)(31,33)(32,34)(35,36);(5,7,10,11)(6,8,12,9)(14,17)(18,20)(22,27)(23,31)(29,32)(30,36);(1,2,4,3)(5,8,10,9)(6,7,12,11)(13,14)(15,18)(16,17)(19,20)(21,23,26,31)(22,28,27,33)(24,30,34,36)(25,32,35,29);(5,6)(7,8)(9,11)(10,12)(13,15,16,19)(14,18,17,20)(21,24)(22,29)(23,30)(25,28)(26,34)(27,32)(31,36)(33,35);(1,3,4,2)(5,9,10,8)(6,11,12,7)(21,25)(22,30)(23,32)(24,33)(26,35)(27,36)(28,34)(29,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1186387);